Overview
Order of operations within math problems gives students a set of rules to follow in order to get the same answer. In the standard order of operations, expressions within parentheses are worked first, then exponents, then multiplication, division, addition, and subtraction.
Conventional Order
Suppose the problem is as simple as 85 +9 X 2. Without an order of operations, one person might calculate the problem as (85 +9) X 2 or 94 X 2 = 188, while another might calculate it as 85 + (9 X 2) or 85 + 18 =103. Similarly, does 2 X 34 mean 64, or 1296, or does it mean to work the 34 and then multiply by 2, or 81 X 2 = 162?
Ensuring Solutions
By convention, exponents are solved first, then multiplications or divisions, from left to right, then additions or subtractions, from left to right. Therefore, the correct answer to 85 + 9 X 2 is not 188, but 103. There are no exponents in the problem, but there is a multiplication, 9 X 2, equaling 18. Working from left to right, the last step is adding 85 + 18, to equal 103. Similarly, in 2 X 34, the exponent is worked first, then the multiplication.
Parentheses
Parentheses in mathematical sentences take priority over all other operations, so they are solved first, from left to right. Suppose the problem really were (85 +9) X 2. In that case, the correct answer would be to solve within the parentheses first, then the multiplication. Similarly, (4x)3 does not mean the same as 4x3. Within the order of operations, if x equals 2, (4x)3 would equal 83 or 512. If x equals 2, 4x3 would equal 4 X 23 or 32.
Types of Calculators and Order of Operations
When they are buying school supplies, students may wonder why their teachers specify certain types of calculators. That is because scientific calculators follow the algebraic order of operations, while other calculators do not. A scientific calculator will calculate the problem 85 +9 X 2 by doing the multiplication first, coming up with the answer 103, while a regular calculator will do the operations from left to right only, answering 188.
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